The local behavior of principal and chordal principal cluster sets

Abstract:

Let K be the unit circle, and let f be a function whose domain is the open unit disk and whose range is a subset of the Riemann sphere. We define a set, called the boundary principal cluster set of f at ${\zeta _0} \in K$, which characterizes the behavior of the principal cluster sets of f at points $\zeta \in K$ which are near ${\zeta _0}$ and distinct from ${\zeta _0}$. It is shown that if f is continuous, then the principal and boundary principal cluster sets of f at ${\zeta _0}$ are equal for nearly every point ${\zeta _0} \in K$. A similar result holds for chordal principal cluster sets. Examples are provided that indicate directions in which the result cannot be improved. Some results concerning points that are accessible through sets which are unions of arcs are also presented.